Weighted Coordinates Poset Block Codes

Abstract

Given $[n]=\{1,2,\ldots,n\}$, a partial order $\preceq$ on $[n]$, a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i) = k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, the direct sum $ \mathbb{F}_{q}^{k_1} \oplus \mathbb{F}_{q}^{k_2}\oplus \ldots \oplus \mathbb{F}_{q}^{k_n} $ of $ \mathbb{F}_q^N $, and a weight function $w$ on $ \mathbb{F}_q $, we define a poset block metric $d_{(P,w,\pi)}$ on $\mathbb{F}_{q}^{N}$ based on the poset $P=([n],\preceq)$. The metric $d_{(P,w,\pi)}$ is said to be weighted coordinates poset block metric ($(P,w,\pi)$-metric). It extends the weighted coordinates poset metric ($(P,w)$-metric) introduced by L. Panek and J. A. Pinheiro and generalizes the poset block metric ($(P,\pi)$-metric) introduced by M. M. S. Alves et al. We determine the complete weight distribution of a $(P,w,\pi)$-space, thereby obtaining it for $(P,w)$-space, $(P,\pi)$-space, $\pi$-space, and $P$-space as special cases. We obtain the Singleton bound for $(P,w,\pi)$-codes and for $(P,w)$-codes as well. In particular, we re-obtain the Singleton bound for any code with respect to $(P,\pi)$-metric and $P$-metric. Moreover, packing radius and Singleton bound for NRT block codes are found.